The latter question was asked in the particular case when \(a(n)\)  are real in [1]}. In the case when \(f\)  is a newform of integral weight without complex multiplication (CM), the celebrate Sato-Tate conjecture suggests that no matter how we slice the plane with a straight line going through the origin, the proportion of primes for which the \(a(p)\)  lies in the same half-plane is equal to the half of the proportion of primes for which the \(a(p)\)  are not on the line. The questions that then naturally arise are: whether this is still true for newforms with CM? Can we infer similar results for \(a(n)\)  when \(n\)  runs through natural integers? Numerical calculations seem to suggest that the answer is positive.
